Theoretical analysis reveals non-invertible symmetries in the Monster CFT via parafermionization, indicating that defect McKay-Thompson series exhibit congruence subgroup invariance.
A bstract We study the parafermionization of the Monster CFT with respect to its ℤ pA subgroups, with p an odd prime. Under certain assumptions, we show that the parafermionization is equal to a non-invertible gauging of P(p)× P(p) P p × P p ∨ , where P(p) P p is the theory of ℤ p -parafermions and P(p) P p ∨ is an appropriate dual theory, with global symmetry characterized by the centralizer of ℤ pA . By tracking the symmetries of P(p)× P(p) P p × P p ∨ through the non-invertible gauging, we argue that the diagonal Monster CFT has Rep(so(3)ₚ) Rep(so(3)ₚ)ᵒᵖ Rep so 3 p ⊠ Rep so 3 p op symmetry, and hence that the holomorphic Monster theory has symmetry Rep(so(3)ₚ) Rep so 3 p . We then compute the defect McKay-Thompson series associated to these symmetries, and prove that their invariance subgroups are Γ 1 ( p + 2).
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Honda et al. (2026) studied this question.
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